The number of parameters in a standard multinomial choice model increases linearly with the number of choice alternatives and number of explanatory variables. Since many modern applications involve large choice sets with categorical explanatory variables, which enter the model as large sets of binary dummies, the number of parameters easily approaches the sample size. This paper proposes methods for data-driven parameter clustering over outcome categories and explanatory dummy categories in a multinomial probit setting. A Dirichlet process mixture encourages parameters to cluster over the categories, which favours a parsimonious model specification without a priori imposing model restrictions. Simulation studies and an application to a dataset of holiday destinations show a decrease in parameter uncertainty and an enhancement of the parameter interpretability, relative to a standard multinomial choice model.